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The dynamical dielectric function of two dimensional graphene at arbitrarywave vector $q$ and frequency $\omega$, $\epsilon(q,\omega)$, is calculated inthe self-consistent field approximation. The results are used to find thedispersion of the plasmon mode and the electrostatic screening of the Coulombinteraction in 2D graphene layer within the random phase approximation. At longwavelengths ($q\to 0$) the plasmon dispersion shows the local classicalbehavior $\omega_{cl} = \omega_0 \sqrt{q}$, but the density dependence of theplasma frequency ($\omega_0 \propto n^{1/4}$) is different from the usual 2Delectron system ($\omega_0 \propto n^{1/2}$). The wave vector dependent plasmondispersion and the static screening function show very different behavior thanthe usual 2D case.Comment: 6 pages, 3 figure

Dielectric function, screening, and plasmons in two-dimensional graphene